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=== Many 12edo temperaments ===
=== Many 12edo temperaments ===
Now, let's consider 12edo, not as a 3-limit temperament, but as a [[5-limit]] temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is {{val| 12 19 28 }}. It's important to keep in mind that this is, technically speaking, a ''different'' regular temperament than {{val| 12 19 }}, even though they would both be referred to as "12-tone equal temperament" in common parlance.
今度は3リミットではなく5リミットテンペラメントとしての12平均律を考える。このテンペラメントは全ての3リミット純正音程を上記と同じようにマップするが、それだけではなくそれ以外の5リミット純正音程も扱う。そのマッピング行列は {{val| 12 19 28 }} となる。技術的に言うとこれは {{val| 12 19 }} とは''異なる''レギュラーテンペラメントであるが、一般的な用語としてはどちらも「12平均律」と呼ばれる。


Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D♯". This choice results in two different mappings, {{val| 12 19 28 34 41 }} and {{val| 12 19 28 34 42 }}. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.
さらに11リミットの12平均律を考える。そのマッピング行列は？ それは 11/8 を「だいぶシャープなD」と考えるか「だいぶフラットなD♯」と考えるかによって変わる。マッピングはそれぞれ  {{val| 12 19 28 34 41 }} と {{val| 12 19 28 34 42 }} となる。後者のほうがより正確な 11/8 を持つが、前者は 12/11 を含むいくつかの音程がより正確になる。RTT流に言えば、オクターブを12等分する11リミットテンペラメントは2個あるということである。「11リミット12平均律」という言い方ではマッピングを特定できていなく、つまり特定のテンペラメントを指し示せていない。


Strictly speaking, "5-limit 12edo" or even "3-limit 12edo" are also ambiguous, because {{val| 12 19 27 }}, for example, is a valid temperament even though it's much less accurate than {{val| 12 19 28 }}. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.
厳密に言えば、「5リミット12平均律」や「3リミット12平均律」ですら同様に曖昧である、なぜなら例えば {{val| 12 19 27 }} だって有効なテンペラメントだからである（{{val| 12 19 28 }} よりずっと不正確ではあるが）。このテンペラメントでは 5/4 が12edoの3ステップ（300 セント）で表される。もちろん実用上は高リミットにならないとこの曖昧性は問題にならない。


== Linear temperament mappings ==
== Linear temperament mappings ==
Now let's consider a temperament that does not consist of a single chain of equally spaced notes. For example, consider conventional music notation ''without'' enharmonic equivalence. Every note of this system can be expressed as some combination of octaves and perfect fourths. For example:
1本の等間隔ピッチのチェーンではないテンペラメントを考える。例として異名同音なし（C♯とD♭をあくまで別物とする）の伝統的記法を考える。全ての音高はオクターブと完全4度の組み合わせとして記述できる。例えば


* E5 = A440 + 1 octave − 1 perfect fourths
* E5 = A440 + 1 octave − 1 perfect fourths
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* A♯ = A440 + 4 octaves − 7 perfect fourths
* A♯ = A440 + 4 octaves − 7 perfect fourths


In other words, every note can be represented as an ordered pair of integers {{nowrap|(''x'', ''y'')}} where ''x'' is the number of octaves from A440 (positive is up, negative is down), and ''y'' is the number of perfect fourths.
言い換えると、全ての音高は整数の組 (''x'', ''y'') で表され、''x'' がオクターブの個数（負なら下がるほう）、''y'' が完全4度の個数である。
 
== Temperamental rank ==
A temperament's "rank" denotes how many independent chains of generators exist within the temperament. This is a mathematical term that's borrowed from the fields of group theory and linear algebra. It can also be viewed as the dimension of the temperament's mapping.
 
For example:
# An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator.
# Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank-2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains.
# Temperaments which consist of three generators, or more commonly a period and two generators, are rank-3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like).
# 7-limit JI would be rank-4, 11-limit JI would be rank-5, 13-limit JI would be rank-6, etc.
 
A single [[val]] in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank 2 or higher, we simply need to use more than one val. In general, the number of vals that it requires to fully map a temperament is equal to the temperament's rank.
 
== Example ==
At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:
 
{{val| a b c }} – period
 
{{val| d e f }} – generator
 
The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.
 
When mapping a prime in JI onto a rank 2 temperament, one must think about how many steps of each type of generator it takes to reach the final tempered prime interval. For an example, we'll look at meantone temperament, and we'll start by mapping 2/1. We'll assume that the period is 2/1, and the generator is 3/2. 2/1 maps to one period and zero generators, and hence we arrive at
 
{{val| 1 _ _ }}
 
{{val| 0 _ _ }}
 
3/1 is slightly more complicated – it requires one step along the 2/1 period chain, plus one step along the 3/2 generator chain, to get to 3/1. This is represented by the following mapping:
 
{{val| 1 1 _ }}
 
{{val| 0 1 _ }}
 
5/1 is simpler—we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:
 
{{val| 1 1 0 }}
 
{{val| 0 1 4 }}
 
This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.
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